Professor Hofmann's current research lies at the interface of harmonic analysis, partial differential equations, and geometric measure theory. In particular, using techniques of harmonic analysis, he studies the interaction between the geometry of the boundary of a domain, and the behavior of solutions of partial differential equations in the domain. His earlier work treated the theory of singular integrals and square functions, and their applications to partial differential equations.

(with A. McIntosh) The solution of the Kato problem in two dimensions, Proceedings of The Conference on Harmonic Analysis and PDE held in El Escorial, Spain in July 2000, Publ. Mat. Vol. extra, 2002 pp. 143-160.